Ray class field extensions of real quadratic fields and solvability of congruences
type: Item
instance of: scholarly article
title: Ray class field extensions of real quadratic fields and solvability of congruences
zbMATH Open document ID: 0579.12007
DOI: 10.1016/0022-314X(85)90021-6
author: Paul B. Massell
published in: Journal of Number Theory
publication date: 1985-01-01T00:00:00Z
review text: Let \(K={\mathbb{Q}}(\sqrt{d})\) be a real quadratic field with class number one and q a rational prime which splits in K, \(q\cong {\mathfrak pp}'.\) The ray class fields (quadratic extensions) of K with conductor \(f| (4){\mathfrak p}\infty_ 1\infty_ 2\) have been constructed and have been used for proving solvability criteria for congruences of biquadratic polynomials mod q. For example, if \(d=11\), than \(\chi^ 4-26\chi^ 2- 7\equiv 0\) mod q is solvable in \({\mathbb{Z}}\) if and only if \(q\equiv 1 mod 4\) and \(a\mp 5b\equiv 1,2,4\) mod 7 where \(q=a^ 2-11b^ 2\), \(a,b>0\), or \(q\equiv 3 mod 4\) and \(\pm a+5b\equiv 1,2,4\) mod 7, where \(q=-a^ 2+11b^ 2\), \(q\neq 7\), \(a,b>0\).
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11A07
zbMATH DE Number: 3927040
zbMATH Keywords: class field theory
zbMATH Keywords: real quadratic field
zbMATH Keywords: class number one
zbMATH Keywords: ray class fields
zbMATH Keywords: solvability criteria
zbMATH Keywords: congruences
zbMATH Keywords: biquadratic polynomials
reviewed by: Jannis A. Antoniadis
MaRDI profile type: Publication
cites work: Note on a Quartic Congruence
cites work: Q3929826
cites work: Signatures of units and congruences (mod 4) in certain real quadratic fields. II.
full work available at URL: 0022-314x(85)90021-6
OpenAlex ID: W2094293452
description: scientific article; zbMATH DE number 3927040